Frobenius conjugacy class
E790517
A Frobenius conjugacy class is the set of all conjugates of a Frobenius element in a Galois group, encapsulating how a prime ideal splits in a given Galois extension.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Frobenius conjugacy class canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T9297040 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Frobenius conjugacy class Context triple: [Chebotarev density theorem, usesConcept, Frobenius conjugacy class]
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A.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
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B.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
C.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
D.
orbit-stabilizer theorem
The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
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E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Frobenius conjugacy class Target entity description: A Frobenius conjugacy class is the set of all conjugates of a Frobenius element in a Galois group, encapsulating how a prime ideal splits in a given Galois extension.
-
A.
Furtwängler’s theorem in class field theory
Furtwängler’s theorem in class field theory is a fundamental result in algebraic number theory that refines the principal ideal theorem by describing how ideal classes capitulate (become principal) in certain abelian extensions of number fields.
-
B.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
-
C.
Algebraic Groups and Class Fields
"Algebraic Groups and Class Fields" is a influential mathematical monograph that develops the deep connections between algebraic group theory and class field theory within number theory and arithmetic geometry.
-
D.
orbit-stabilizer theorem
The orbit-stabilizer theorem is a fundamental result in group theory that relates the size of a group acting on a set to the sizes of the orbit of an element and its stabilizer subgroup.
-
E.
Deligne–Lusztig theory
Deligne–Lusztig theory is a framework in algebraic geometry and representation theory that constructs and studies representations of finite groups of Lie type using varieties defined over finite fields.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
conjugacy class
ⓘ
mathematical concept ⓘ object in algebraic number theory ⓘ |
| appearsIn |
Euler products of Artin L-functions
ⓘ
definition of Artin reciprocity map ⓘ statement of the Chebotarev density theorem ⓘ |
| arisesFrom |
Frobenius automorphism at a prime
ⓘ
linked to:
Frobenius element
|
| associatedTo | Frobenius element ⓘ |
| characterizedBy |
action on residue fields at primes
ⓘ
cycle structure on roots of polynomials modulo primes ⓘ |
| definedFor | unramified prime ideals ⓘ |
| definedInContextOf |
Galois representations
ⓘ
Galois theory ⓘ algebraic number theory ⓘ |
| dependsOn | choice of prime ideal above a rational prime up to conjugacy ⓘ |
| domain |
Galois extension of function fields
ⓘ
Galois extension of global fields ⓘ Galois extension of number fields ⓘ |
| encodes |
cycle type of Frobenius action on embeddings
ⓘ
decomposition of primes in number fields ⓘ splitting behavior of a prime ideal in a Galois extension ⓘ |
| generalizes | classical Frobenius element in finite field extensions ⓘ |
| hasDefiningGroup | Galois group ⓘ |
| hasElementType | automorphisms of a field extension ⓘ |
| hasNotation |
Frob_p
ⓘ
Frob_𝔭 ⓘ |
| hasProperty |
conjugacy-invariant subset of a Galois group
ⓘ
finite subset when the Galois group is finite ⓘ |
| independentOf | choice of prime above a given unramified prime (up to conjugacy) ⓘ |
| isInvariantUnder | inner automorphisms of the Galois group ⓘ |
| isSetOf | conjugates of a Frobenius element ⓘ |
| playsRoleIn |
classification of primes by splitting type
ⓘ
comparison of different Galois representations via traces of Frobenius ⓘ equidistribution of primes in Galois extensions ⓘ |
| relatedTo |
Artin symbol
ⓘ
linked to:
Frobenius element
Frobenius automorphism ⓘ
linked to:
Frobenius endomorphism
decomposition group ⓘ inertia group ⓘ |
| undefinedFor | ramified prime ideals without additional choices ⓘ |
| usedIn |
Artin L-functions
ⓘ
Chebotarev density theorem ⓘ Langlands program ⓘ Sato–Tate type equidistribution statements ⓘ Serre’s modularity conjecture ⓘ
linked to:
Serre’s conjecture on Galois representations
proofs of equidistribution results for primes ⓘ study of Galois representations of number fields ⓘ |
| usedToDefine |
Artin conductor contributions at unramified primes
ⓘ
local factors of L-functions ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Frobenius conjugacy class Description of subject: A Frobenius conjugacy class is the set of all conjugates of a Frobenius element in a Galois group, encapsulating how a prime ideal splits in a given Galois extension.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.